Sampling from a Log-Concave Distribution with Projected Langevin Monte Carlo
We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected stochastic gradient descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Mark...
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| Published in | Discrete & computational geometry Vol. 59; no. 4; pp. 757 - 783 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.06.2018
Springer Nature B.V Springer Verlag |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0179-5376 1432-0444 |
| DOI | 10.1007/s00454-018-9992-1 |
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| Abstract | We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected stochastic gradient descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in
O
~
(
n
7
)
steps (where
n
is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of
O
~
(
n
4
)
was proved by Lovász and Vempala. |
|---|---|
| AbstractList | We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected stochastic gradient descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in O~(n7) steps (where n is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of O~(n4) was proved by Lovász and Vempala. We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in O(n 7) steps (where n is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of O(n 4) was proved by Lovász and Vempala. We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected stochastic gradient descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in O ~ ( n 7 ) steps (where n is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of O ~ ( n 4 ) was proved by Lovász and Vempala. |
| Author | Bubeck, Sébastien Lehec, Joseph Eldan, Ronen |
| Author_xml | – sequence: 1 givenname: Sébastien surname: Bubeck fullname: Bubeck, Sébastien organization: Microsoft Research – sequence: 2 givenname: Ronen orcidid: 0000-0002-0678-741X surname: Eldan fullname: Eldan, Ronen email: roneneldan@gmail.com organization: Faculty of Mathematics and Computer Science, The Weizmann Institute of Science – sequence: 3 givenname: Joseph surname: Lehec fullname: Lehec, Joseph organization: CEREMADE, Université Paris-Dauphine |
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| Cites_doi | 10.1214/aop/1176992442 10.1145/2746539.2746563 10.1002/rsa.20135 10.1007/978-3-642-20212-4 10.1214/aoms/1177729586 10.1137/S009753970544727X 10.1287/moor.1110.0519 10.32917/hmj/1206135203 10.1137/1106035 10.1145/102782.102783 10.1111/rssb.12183 10.1137/0324039 10.1214/11-AIHP464 10.2307/3318418 |
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| Keywords | 68W20 Langevin Monte Carlo Rapidly-mixing random walks Log-concave measures Sampling and optimization 68W25 47N10 |
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| References | Cousins, B., Vempala, S.: Bypassing KLS: Gaussian cooling and an O∗(n3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm O}}^*(n^3)$$\end{document} volume algorithm. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC’15), pp. 539–548. ACM, New York (2015) LovászLVempalaSHit-and-run from a cornerSIAM J. Comput.20063549851005220373510.1137/S009753970544727X1103.52002 Welling, M., Teh, Y.W.: Bayesian learning via stochastic gradient Langevin dynamics. In: Proceedings of the 28th International Conference on International Conference on Machine Learning (ICML’11), pp. 681–688. Omnipress (2011) KannanRNarayananHRandom walks on polytopes and an affine interior point method for linear programmingMath. Oper. Res.201237120289114410.1287/moor.1110.05191243.65033 Dalalyan, A.S.: Theoretical guarantees for approximate sampling from a smooth and log-concave densities. J. R. Stat. Soc. Stat. Methodol. Ser. B. https://doi.org/10.1111/rssb.12183 NemirovskyASYudinDBProblem Complexity and Method Efficiency in Optimization.1983New YorkWiley LovászLVempalaSThe geometry of logconcave functions and sampling algorithmsRandom Struct. Algorithm.2007303307358230962110.1002/rsa.201351122.65012 Bach, F., Moulines, E.: Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm O}}(1/n)$$\end{document}. In: Proceedings of the 26th International Conference on Neural Information Processing Systems (NIPS’13), vol. 1, pp. 773–781. Curran Associates (2013) LindvallTRogersLCGCoupling of multidimensional diffusions by reflectionAnn. Probab.198614386087284158810.1214/aop/11769924420593.60076 LehecJRepresentation formula for the entropy and functional inequalitiesAnn. Inst. Henri Poincaré Probab. Stat.2013493885899311243810.1214/11-AIHP4641279.39011 RobbinsHMonroSA stochastic approximation methodAnn. Math. Stat.1951224004074266810.1214/aoms/11777295860054.05901 Pflug, G.Ch.: Stochastic minimization with constant step-size: asymptotic laws. SIAM J. Control Optim. 24(4), 655–666 (1986) LedouxMTalagrandMProbability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete1991BerlinSpringer DyerMFriezeAKannanRA random polynomial-time algorithm for approximating the volume of convex bodiesJ. Assoc. Comput. Mach.1991381117109591610.1145/102782.1027830799.68107 Ahn, S., Korattikara, A., Welling, M.: Bayesian posterior sampling via stochastic gradient Fisher scoring. In: Proceedings of the 29th International Conference on Machine Learning (ICML’12), pp. 782–846. IMLS (2012) LevinDAPeresYWilmerELMarkov Chains and Mixing Times2009ProvidenceAmerican Mathematical Society1160.60001 RobertsGOTweedieRLExponential convergence of Langevin distributions and their discrete approximationsBernoulli199624341363144027310.2307/33184180870.60027 TanakaHStochastic differential equations with reflecting boundary condition in convex regionsHiroshima Math. J.1979911631775293320423.60055 SkorokhodAVStochastic equations for diffusion processes in a bounded regionTheory Probab. Appl.196163264274163099910.1137/11060350215.53501 R Kannan (9992_CR6) 2012; 37 M Ledoux (9992_CR7) 1991 AS Nemirovsky (9992_CR13) 1983 M Dyer (9992_CR5) 1991; 38 9992_CR4 DA Levin (9992_CR9) 2009 9992_CR3 L Lovász (9992_CR12) 2007; 30 9992_CR2 9992_CR1 H Robbins (9992_CR15) 1951; 22 L Lovász (9992_CR11) 2006; 35 H Tanaka (9992_CR18) 1979; 9 9992_CR19 GO Roberts (9992_CR16) 1996; 2 9992_CR14 T Lindvall (9992_CR10) 1986; 14 J Lehec (9992_CR8) 2013; 49 AV Skorokhod (9992_CR17) 1961; 6 |
| References_xml | – reference: SkorokhodAVStochastic equations for diffusion processes in a bounded regionTheory Probab. Appl.196163264274163099910.1137/11060350215.53501 – reference: LevinDAPeresYWilmerELMarkov Chains and Mixing Times2009ProvidenceAmerican Mathematical Society1160.60001 – reference: LehecJRepresentation formula for the entropy and functional inequalitiesAnn. Inst. Henri Poincaré Probab. Stat.2013493885899311243810.1214/11-AIHP4641279.39011 – reference: Welling, M., Teh, Y.W.: Bayesian learning via stochastic gradient Langevin dynamics. In: Proceedings of the 28th International Conference on International Conference on Machine Learning (ICML’11), pp. 681–688. Omnipress (2011) – reference: Dalalyan, A.S.: Theoretical guarantees for approximate sampling from a smooth and log-concave densities. J. R. Stat. Soc. Stat. Methodol. Ser. B. https://doi.org/10.1111/rssb.12183 – reference: Bach, F., Moulines, E.: Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm O}}(1/n)$$\end{document}. In: Proceedings of the 26th International Conference on Neural Information Processing Systems (NIPS’13), vol. 1, pp. 773–781. Curran Associates (2013) – reference: KannanRNarayananHRandom walks on polytopes and an affine interior point method for linear programmingMath. Oper. Res.201237120289114410.1287/moor.1110.05191243.65033 – reference: RobertsGOTweedieRLExponential convergence of Langevin distributions and their discrete approximationsBernoulli199624341363144027310.2307/33184180870.60027 – reference: NemirovskyASYudinDBProblem Complexity and Method Efficiency in Optimization.1983New YorkWiley – reference: Ahn, S., Korattikara, A., Welling, M.: Bayesian posterior sampling via stochastic gradient Fisher scoring. In: Proceedings of the 29th International Conference on Machine Learning (ICML’12), pp. 782–846. IMLS (2012) – reference: LovászLVempalaSHit-and-run from a cornerSIAM J. Comput.20063549851005220373510.1137/S009753970544727X1103.52002 – reference: RobbinsHMonroSA stochastic approximation methodAnn. Math. Stat.1951224004074266810.1214/aoms/11777295860054.05901 – reference: LovászLVempalaSThe geometry of logconcave functions and sampling algorithmsRandom Struct. Algorithm.2007303307358230962110.1002/rsa.201351122.65012 – reference: Cousins, B., Vempala, S.: Bypassing KLS: Gaussian cooling and an O∗(n3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm O}}^*(n^3)$$\end{document} volume algorithm. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC’15), pp. 539–548. ACM, New York (2015) – reference: LindvallTRogersLCGCoupling of multidimensional diffusions by reflectionAnn. Probab.198614386087284158810.1214/aop/11769924420593.60076 – reference: LedouxMTalagrandMProbability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete1991BerlinSpringer – reference: Pflug, G.Ch.: Stochastic minimization with constant step-size: asymptotic laws. SIAM J. Control Optim. 24(4), 655–666 (1986) – reference: TanakaHStochastic differential equations with reflecting boundary condition in convex regionsHiroshima Math. J.1979911631775293320423.60055 – reference: DyerMFriezeAKannanRA random polynomial-time algorithm for approximating the volume of convex bodiesJ. Assoc. Comput. Mach.1991381117109591610.1145/102782.1027830799.68107 – volume: 14 start-page: 860 issue: 3 year: 1986 ident: 9992_CR10 publication-title: Ann. Probab. doi: 10.1214/aop/1176992442 – ident: 9992_CR3 doi: 10.1145/2746539.2746563 – volume-title: Problem Complexity and Method Efficiency in Optimization. year: 1983 ident: 9992_CR13 – volume: 30 start-page: 307 issue: 3 year: 2007 ident: 9992_CR12 publication-title: Random Struct. Algorithm. doi: 10.1002/rsa.20135 – volume-title: Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete year: 1991 ident: 9992_CR7 doi: 10.1007/978-3-642-20212-4 – volume: 22 start-page: 400 year: 1951 ident: 9992_CR15 publication-title: Ann. Math. Stat. doi: 10.1214/aoms/1177729586 – volume: 35 start-page: 985 issue: 4 year: 2006 ident: 9992_CR11 publication-title: SIAM J. Comput. doi: 10.1137/S009753970544727X – volume: 37 start-page: 1 year: 2012 ident: 9992_CR6 publication-title: Math. Oper. Res. doi: 10.1287/moor.1110.0519 – volume: 9 start-page: 163 issue: 1 year: 1979 ident: 9992_CR18 publication-title: Hiroshima Math. J. doi: 10.32917/hmj/1206135203 – volume: 6 start-page: 264 issue: 3 year: 1961 ident: 9992_CR17 publication-title: Theory Probab. Appl. doi: 10.1137/1106035 – ident: 9992_CR1 – ident: 9992_CR2 – volume: 38 start-page: 1 issue: 1 year: 1991 ident: 9992_CR5 publication-title: J. Assoc. Comput. Mach. doi: 10.1145/102782.102783 – volume-title: Markov Chains and Mixing Times year: 2009 ident: 9992_CR9 – ident: 9992_CR4 doi: 10.1111/rssb.12183 – ident: 9992_CR14 doi: 10.1137/0324039 – ident: 9992_CR19 – volume: 49 start-page: 885 issue: 3 year: 2013 ident: 9992_CR8 publication-title: Ann. Inst. Henri Poincaré Probab. Stat. doi: 10.1214/11-AIHP464 – volume: 2 start-page: 341 issue: 4 year: 1996 ident: 9992_CR16 publication-title: Bernoulli doi: 10.2307/3318418 |
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| Snippet | We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected stochastic gradient descent (SGD).... We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD).... |
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| SubjectTerms | Combinatorics Computational Mathematics and Numerical Analysis Markov chains Mathematics Mathematics and Statistics Monte Carlo simulation Probability |
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| Title | Sampling from a Log-Concave Distribution with Projected Langevin Monte Carlo |
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